Arithmetic progressions of three prime numbers with two primes of the form $\mathbf{p=x^2+y^2+1}$

TL;DR

This paper proves the existence of infinitely many three-term arithmetic progressions of primes where two primes are of the form x^2 + y^2 + 1, expanding understanding of prime distributions with specific algebraic properties.

Contribution

It establishes the infinite occurrence of such prime triplets with two primes of the specified quadratic form, a novel result in prime number theory.

Findings

Infinitely many arithmetic progressions of three primes exist with two primes of the form x^2 + y^2 + 1.

The primes p_1 and p_2 in these progressions satisfy p_3=2p_2-p_1.

The primes p_1 and p_2 are of the form x^2 + y^2 + 1.

Abstract

In the present paper we prove that there exist infinitely many arithmetic progressions of three different primes $p_1,p_2,p_3=2p_2-p_1$ such that $p_1=x_1^2 + y_1^2 +1$, $p_2=x_2^2 + y_2^2 +1$.